Logic gates and digital circuits

Explain what a digital signal is and how it differs from an analogue signal, giving an everyday example of each.

Classify these signals as analogue or digital and justify your answer: the volume knob on an audio dial, a light switch, the temperature shown on a mercury thermometer, and the status of a door (open or closed).

How many rows does the truth table of a circuit with two inputs have? And with three? Deduce the general rule for $n$ inputs and test it with four.

Write the complete truth table of a two-input OR gate and explain its behaviour with an example: two doorbells, one on each floor, that sound the same chime.

If an AND gate receives A = 1 and B = 0, what is its output value? What if it were an OR gate with the same inputs? Reason each answer.

Write the complete truth table of a two-input AND gate and describe a real-world situation that works with the same logic, for example in gate security or account security.

Write the truth table of a NOT gate and explain its behaviour with an example: an indicator light that comes on when a machine is NOT running.

Explain what happens when two NOT gates are connected one after the other and prove it by writing the truth table for the combination. What is this equivalence called in logic?

In the Technology workshop you only have 7400 integrated circuits available, which contain exclusively two-input NAND gates. You need to implement the acoustic warning system for a machine, whose simplified logic function is S = A + NOT B.

1. a) Apply De Morgan's laws to transform this function into an equivalent expression that uses exclusively NAND operations and negations.
2. b) Demonstrate how to construct a NOT gate using only a NAND gate and write its truth table.
3. c) Draw or describe step by step the resulting circuit diagram using only two-input NAND gates, indicating the total number of gates required and explaining whether any of them can be simplified.

The figure shows a circuit with two gates: inputs A and B go into an AND gate, and its output along with C goes into an OR gate. Write the complete truth table of the circuit and calculate the value of S for the specified case.

A fan must switch on ($F = 1$) only when it is hot ($C = 1$) and the user allows it ($P = 1$), but never when the window is open ($V = 1$). A classmate has built the control with two logic gates: an AND gate combining $C$ and $P$, and an OR gate combining that result with $V$, as shown in the diagram. When tested in the workshop, the fan switches on whenever the window is open, even in the middle of winter!

1. a) Draw up the truth table of your classmate's circuit and point out the cases where it does not meet the specification.
2. b) Explain the conceptual mistake of using the OR gate with $V$.
3. c) Design the correct circuit: write the simplified logic function and state the gates needed.

A technician is designing the control system for a pedestrian green light ($V$) at a tram track crossing. The circuit has three binary input signals:

• Tram sensor: $T = 1$ if a tram is approaching and $T = 0$ if the track is clear.

• Pedestrian push-button: $P = 1$ if there are waiting pedestrians who have pressed the button and $P = 0$ if it has not been pressed.

• Night mode: $N = 1$ during the night and $N = 0$ during the day.

The safety specification requires that the green pedestrian light can only turn on ($V = 1$) when no tram is approaching ($T = 0$) and at least one of the following two conditions is met: a pedestrian has pressed the button ($P = 1$) or the system is in night mode ($N = 1$, to allow periodic automatic crossing). Under no circumstances should it turn on if the tram is coming ($T = 1$).

The technician hastily proposes the logic function: $V = (P \cdot \overline{T}) + N$.

1. a) Construct the truth table for the technician's proposal and point out which input combinations will cause a risk of an imminent collision (green pedestrian light with an approaching tram).

1. b) Correct the logic function so that it strictly complies with all safety requirements and draw the circuit schematic using basic logic gates (NOT, AND, OR).

1. c) Analyse what would happen if sensor $T$ fails and the circuit permanently interprets a low level ($T = 0$): is the system fault-tolerant to this failure or does it create a hazardous situation? Justify your answer.